- FindStat

Identifier

St000506: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        [1]=>0
[2]=>0
[1,1]=>1
[3]=>0
[2,1]=>1
[1,1,1]=>0
[4]=>0
[3,1]=>1
[2,2]=>1
[2,1,1]=>1
[1,1,1,1]=>1
[5]=>0
[4,1]=>1
[3,2]=>2
[3,1,1]=>2
[2,2,1]=>2
[2,1,1,1]=>2
[1,1,1,1,1]=>0
[6]=>0
[5,1]=>1
[4,2]=>3
[4,1,1]=>3
[3,3]=>2
[3,2,1]=>6
[3,1,1,1]=>4
[2,2,2]=>2
[2,2,1,1]=>4
[2,1,1,1,1]=>2
[1,1,1,1,1,1]=>1
[7]=>0
[6,1]=>1
[5,2]=>4
[5,1,1]=>4
[4,3]=>5
[4,2,1]=>12
[4,1,1,1]=>7
[3,3,1]=>8
[3,2,2]=>8
[3,2,1,1]=>14
[3,1,1,1,1]=>6
[2,2,2,1]=>6
[2,2,1,1,1]=>6
[2,1,1,1,1,1]=>3
[1,1,1,1,1,1,1]=>0
[8]=>0
[7,1]=>1
[6,2]=>5
[6,1,1]=>5
[5,3]=>9
[5,2,1]=>20
[5,1,1,1]=>11
[4,4]=>5
[4,3,1]=>25
[4,2,2]=>20
[4,2,1,1]=>33
[4,1,1,1,1]=>13
[3,3,2]=>16
[3,3,1,1]=>22
[3,2,2,1]=>28
[3,2,1,1,1]=>26
[3,1,1,1,1,1]=>9
[2,2,2,2]=>6
[2,2,2,1,1]=>12
[2,2,1,1,1,1]=>9
[2,1,1,1,1,1,1]=>3
[1,1,1,1,1,1,1,1]=>1
[9]=>0
[8,1]=>1
[7,2]=>6
[7,1,1]=>6
[6,3]=>14
[6,2,1]=>30
[6,1,1,1]=>16
[5,4]=>14
[5,3,1]=>54
[5,2,2]=>40
[5,2,1,1]=>64
[5,1,1,1,1]=>24
[4,4,1]=>30
[4,3,2]=>61
[4,3,1,1]=>80
[4,2,2,1]=>81
[4,2,1,1,1]=>72
[4,1,1,1,1,1]=>22
[3,3,3]=>16
[3,3,2,1]=>66
[3,3,1,1,1]=>48
[3,2,2,2]=>34
[3,2,2,1,1]=>66
[3,2,1,1,1,1]=>44
[3,1,1,1,1,1,1]=>12
[2,2,2,2,1]=>18
[2,2,2,1,1,1]=>21
[2,2,1,1,1,1,1]=>12
[2,1,1,1,1,1,1,1]=>4
[1,1,1,1,1,1,1,1,1]=>0
[10]=>0
[9,1]=>1
[8,2]=>7
[8,1,1]=>7
[7,3]=>20
[7,2,1]=>42
[7,1,1,1]=>22
[6,4]=>28
[6,3,1]=>98
[6,2,2]=>70
[6,2,1,1]=>110
[6,1,1,1,1]=>40
[5,5]=>14
[5,4,1]=>98
[5,3,2]=>155
[5,3,1,1]=>198
[5,2,2,1]=>185
[5,2,1,1,1]=>160
[5,1,1,1,1,1]=>46
[4,4,2]=>91
[4,4,1,1]=>110
[4,3,3]=>77
[4,3,2,1]=>288
[4,3,1,1,1]=>200
[4,2,2,2]=>115
[4,2,2,1,1]=>219
[4,2,1,1,1,1]=>138
[4,1,1,1,1,1,1]=>34
[3,3,3,1]=>82
[3,3,2,2]=>100
[3,3,2,1,1]=>180
[3,3,1,1,1,1]=>92
[3,2,2,2,1]=>118
[3,2,2,1,1,1]=>131
[3,2,1,1,1,1,1]=>68
[3,1,1,1,1,1,1,1]=>16
[2,2,2,2,2]=>18
[2,2,2,2,1,1]=>39
[2,2,2,1,1,1,1]=>33
[2,2,1,1,1,1,1,1]=>16
[2,1,1,1,1,1,1,1,1]=>4
[1,1,1,1,1,1,1,1,1,1]=>1
[11]=>0
[10,1]=>1
[9,2]=>8
[9,1,1]=>8
[8,3]=>27
[8,2,1]=>56
[8,1,1,1]=>29
[7,4]=>48
[7,3,1]=>160
[7,2,2]=>112
[7,2,1,1]=>174
[7,1,1,1,1]=>62
[6,5]=>42
[6,4,1]=>224
[6,3,2]=>323
[6,3,1,1]=>406
[6,2,2,1]=>365
[6,2,1,1,1]=>310
[6,1,1,1,1,1]=>86
[5,5,1]=>112
[5,4,2]=>344
[5,4,1,1]=>406
[5,3,3]=>232
[5,3,2,1]=>826
[5,3,1,1,1]=>558
[5,2,2,2]=>300
[5,2,2,1,1]=>564
[5,2,1,1,1,1]=>344
[5,1,1,1,1,1,1]=>80
[4,4,3]=>168
[4,4,2,1]=>489
[4,4,1,1,1]=>310
[4,3,3,1]=>447
[4,3,2,2]=>503
[4,3,2,1,1]=>887
[4,3,1,1,1,1]=>430
[4,2,2,2,1]=>452
[4,2,2,1,1,1]=>488
[4,2,1,1,1,1,1]=>240
[4,1,1,1,1,1,1,1]=>50
[3,3,3,2]=>182
[3,3,3,1,1]=>262
[3,3,2,2,1]=>398
[3,3,2,1,1,1]=>403
[3,3,1,1,1,1,1]=>160
[3,2,2,2,2]=>136
[3,2,2,2,1,1]=>288
[3,2,2,1,1,1,1]=>232
[3,2,1,1,1,1,1,1]=>100
[3,1,1,1,1,1,1,1,1]=>20
[2,2,2,2,2,1]=>57
[2,2,2,2,1,1,1]=>72
[2,2,2,1,1,1,1,1]=>49
[2,2,1,1,1,1,1,1,1]=>20
[2,1,1,1,1,1,1,1,1,1]=>5
[1,1,1,1,1,1,1,1,1,1,1]=>0
[12]=>0
[11,1]=>1
[10,2]=>9
[10,1,1]=>9
[9,3]=>35
[9,2,1]=>72
[9,1,1,1]=>37
[8,4]=>75
[8,3,1]=>243
[8,2,2]=>168
[8,2,1,1]=>259
[8,1,1,1,1]=>91
[7,5]=>90
[7,4,1]=>432
[7,3,2]=>595
[7,3,1,1]=>740
[7,2,2,1]=>651
[7,2,1,1,1]=>546
[7,1,1,1,1,1]=>148
[6,6]=>42
[6,5,1]=>378
[6,4,2]=>891
[6,4,1,1]=>1036
[6,3,3]=>555
[6,3,2,1]=>1920
[6,3,1,1,1]=>1274
[6,2,2,2]=>665
[6,2,2,1,1]=>1239
[6,2,1,1,1,1]=>740
[6,1,1,1,1,1,1]=>166
[5,5,2]=>456
[5,5,1,1]=>518
[5,4,3]=>744
[5,4,2,1]=>2065
[5,4,1,1,1]=>1274
[5,3,3,1]=>1505
[5,3,2,2]=>1629
[5,3,2,1,1]=>2835
[5,3,1,1,1,1]=>1332
[5,2,2,2,1]=>1316
[5,2,2,1,1,1]=>1396
[5,2,1,1,1,1,1]=>664
[5,1,1,1,1,1,1,1]=>130
[4,4,4]=>168
[4,4,3,1]=>1104
[4,4,2,2]=>992
[4,4,2,1,1]=>1686
[4,4,1,1,1,1]=>740
[4,3,3,2]=>1132
[4,3,3,1,1]=>1596
[4,3,2,2,1]=>2240
[4,3,2,1,1,1]=>2208
[4,3,1,1,1,1,1]=>830
[4,2,2,2,2]=>588
[4,2,2,2,1,1]=>1228
[4,2,2,1,1,1,1]=>960
[4,2,1,1,1,1,1,1]=>390
[4,1,1,1,1,1,1,1,1]=>70
[3,3,3,3]=>182
[3,3,3,2,1]=>842
[3,3,3,1,1,1]=>665
[3,3,2,2,2]=>534
[3,3,2,2,1,1]=>1089
[3,3,2,1,1,1,1]=>795
[3,3,1,1,1,1,1,1]=>260
[3,2,2,2,2,1]=>481
[3,2,2,2,1,1,1]=>592
[3,2,2,1,1,1,1,1]=>381
[3,2,1,1,1,1,1,1,1]=>140
[3,1,1,1,1,1,1,1,1,1]=>25
[2,2,2,2,2,2]=>57
[2,2,2,2,2,1,1]=>129
[2,2,2,2,1,1,1,1]=>121
[2,2,2,1,1,1,1,1,1]=>69
[2,2,1,1,1,1,1,1,1,1]=>25
[2,1,1,1,1,1,1,1,1,1,1]=>5
[1,1,1,1,1,1,1,1,1,1,1,1]=>1
                    

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Description

The number of standard desarrangement tableaux of shape equal to the given partition.
A standard desarrangement tableau is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation).
This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also:

St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition.: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition
St000500Eigenvalues of the random-to-random operator acting on the regular representation.: Eigenvalues of the random-to-random operator acting on the regular representation.

Code

def tableau_ascents(t):
    r"""
    The (sorted list) of ascents of the standard tableau `t`.

    An *ascent* of a standard tableau `t` is an entry `i`
    such that `i+1` apears to the right or above `i` in `t`
    (in English notation for tableaux).
    """
    locations = {}
    for (i, row) in enumerate(t):
        for (j, entry) in enumerate(row):
            locations[entry] = (i, j)
    ascents = [t.size()]
    for i in range(1, t.size()):
        # ascent means i+1 appears to the right or above
        x, _ = locations[i]
        u, _ = locations[i+1]
        if u <= x:
            ascents.append(i)
    return sorted(ascents)

def is_desarrangement_tableau(t):
    r"""
    Test whether a tableau is a desarrangement tableau.

    A *desarrangement tableau* is a standard tableau
    whose first ascent is even.
    """
    return min(tableau_ascents(Tableau(t))) % 2 == 0

def statistic(la):
    return len([t for t in StandardTableaux(la) if is_desarrangement_tableau(t)])

Created

May 24, 2016 at 23:10 by Franco Saliola

Updated

Jun 11, 2016 at 01:03 by Martin Rubey